It lists the normal common sense things you would expect as well as an agreement that you will not open carry while on the range. It's that simple. If you cannot comply with the agreement you have signed, you lose your shooting privileges. Remember, you have agreed in writing not to open carry.
There is a valid question as to whether or not the range can legally make that restriction. Common sense or not, it sounds like an "ordinance, regulation, or rule pertaining to firearms", and only the state legislature can make those, per 76-10-500. Unless there's some other statute that grants ranges permission to regulate carry methods, it's an illegal rule.
Unfortunately, the same argument probably applies to other absolutely essential range safety rules, such as ceasing fire when people are going downrange to check their targets. Of course, someone violating *that* rule should probably be arrested and prosecuted for reckless endangerment, so you could argue that it's not so much a range rule as an application of existing law. When the Range Officer calls "cease fire" he's not giving order, he's informing you that people are going to be heading downrange, and choosing to fire would be unsafe and unlawful.
Other rules, such as the "no rapid fire" rule are interesting questions. I wouldn't be surprised that any number of the rules are actually illegal. Personally, I wouldn't press the issue, but someone probably could.